Theorem raleqtrdv 2701

Description: Substitution of equal classes into a restricted universal quantifier. (Contributed by Matthew House, 21-Jul-2025.)

Hypotheses

Ref	Expression
raleqtrdv.1	|- ( ph -> A. x e. A ps )
raleqtrdv.2	|- ( ph -> A = B )

Assertion

Ref	Expression
raleqtrdv	|- ( ph -> A. x e. B ps )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem raleqtrdv

Step	Hyp	Ref	Expression
1		raleqtrdv.1	. 2 |- ( ph -> A. x e. A ps )
2		raleqtrdv.2	. . 3 |- ( ph -> A = B )
3	2	raleqdv 2699	. 2 |- ( ph -> ( A. x e. A ps <-> A. x e. B ps ) )
4	1, 3	mpbid 147	1 |- ( ph -> A. x e. B ps )

Syntax hints:   -> wi 4   = wceq 1364   A.wral 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480

This theorem is referenced by:  znf1o  14207